Atlas › Modular forms › Weight 12 › Level 16 › 16.12.a › 16.12.a.a

Newform orbit 16.12.a.a

$q$ -expansion

$q - 252q^{3} + 4830q^{5} + 16744q^{7} - 113643q^{9} - 534612q^{11} - 577738q^{13} - 1217160q^{15} - 6905934q^{17} - 10661420q^{19} - 4219488q^{21} - 18643272q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}$ . The eigenvalue of $T = T_{3} + T_{5}$ is a root of

$x - 4578$

Atkin–Lehner signs

$p$	$w_p$
2	— ( $2^2 \mid 16$ )

For $p \parallel N$ every form in the orbit has $a_p = -p^{k/2-1} w_p$ .

Traces of Hecke eigenvalues

$p$	2	3	5	7	11	13	17	19	23	29	31	37	41	43	47	53	59	61	67	71	73	79	83	89	97
$\operatorname{tr} a_p$	0	−252	4830	16744	−534612	−577738	−6905934	−10661420	−18643272	128406630	52843168	−182213314	308120442	17125708	−2687348496	−1596055698	5189203740	6956478662	15481826884	−9791485272	1463791322	−38116845680	29335099668	−24992917110	75013568546

$\operatorname{tr} a_n$ for every $n \le 1000$ is in the JSON.

Sato–Tate

The normalized $a_p / 2p^{11/2}$ for the primes $p < 1000$ not dividing 16, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.342 s (one thread) and proven: the dimensions are certified, and the characteristic polynomial of $T$ over $\mathbb{Z}$ is reconstructed by CRT past a rigorous coefficient bound.

The engine's checks

dim M^+ at levels [1, 2, 4, 8, 16]: [2, 4, 7, 13, 24] (certified at ell = 2147483629)
T = [(3, 1), (5, 1)]; h of degree 5 = 1 x 5 (phi(ord eps) x dim S_k^new); 3 primes, 0 rejected

Open in the notebook · JSON · stored data

Recompute it yourself

In Sage mode at sagebrush.space (or npx sagebrush --sage):

newform_orbits(16, 12)[0]   # 16.12.a.a

With the Rust engine (source), in engine/:

cargo run --release -p sagebrush-web --example atlas -- 16:12 1000